Eigenstates, Eigenvalues

Marian J. R. Gilton

Brian P. Anderson

Anthony F. J. Levi

Mark S. Hillery, Vladimír Bužek

We consider several problems that involve finding the eigenvalues and generating the eigenstates of unknown unitary gates. We first examine controlled-$U$ gates that act on qubits and assume that we know the eigenvalues. It is then shown how to use singlet states to produce qubits in the eigenstates of the gate. We then remove the assumption that we know the eigenvalues and show how to both find the eigenvalues and produce qubits in the eigenstates. Finally, we look at the case where the unitary operation acts on qutrits and it has two eigenvalues of $1$ and one of $\ensuremath{-}1.$ We are able to use a singlet state to produce a qutrit in the eigenstate corresponding to the $\ensuremath{-}1$ eigenvalue.

T. Kebukawa

In a system of N spins ordered on a linear chain, an eigenvalue problem for the square and z-component of a total spin operator is solved exactly by using the Bethe ansatz. It is shown that the eigenstates are given by the solutions of coupled equations. For a given eigenvalue of the square of the total spin, the eigenstates for various eigenvalues of the z-component are expressed in a systematic and compact form.

Tomislav P. Živković